Transformations of Functions | Calculus I (2024)

We have seen several cases in which we have added, subtracted, or multiplied constants to form variations of simple functions. In the previous example, for instance, we subtracted 2 from the argument of the function [latex]y=x^2[/latex] to get the function [latex]f(x)=(x-2)^2[/latex]. This subtraction represents a shift of the function [latex]y=x^2[/latex] two units to the right. A shift, horizontally or vertically, is a type of transformation of a function. Other transformations include horizontal and vertical scalings, and reflections about the axes.

Vertical Shift

A vertical shift of a function occurs if we add or subtract the same constant to each output [latex]y[/latex]. For [latex]c>0[/latex], the graph of [latex]f(x)+c[/latex] is a shift of the graph of [latex]f(x)[/latex] up [latex]c[/latex] units, whereas the graph of [latex]f(x)-c[/latex] is a shift of the graph of [latex]f(x)[/latex] down [latex]c[/latex] units. For example, the graph of the function [latex]f(x)=x^3+4[/latex] is the graph of [latex]y=x^3[/latex] shifted up 4 units; the graph of the function [latex]f(x)=x^3-4[/latex] is the graph of [latex]y=x^3[/latex] shifted down 4 units (Figure 15).

Horizontal Shift

A horizontal shift of a function occurs if we add or subtract the same constant to each input [latex]x[/latex]. For [latex]c>0[/latex], the graph of [latex]f(x+c)[/latex] is a shift of the graph of [latex]f(x)[/latex] to the left [latex]c[/latex] units; the graph of [latex]f(x-c)[/latex] is a shift of the graph of [latex]f(x)[/latex] to the right [latex]c[/latex] units. Why does the graph shift left when adding a constant and shift right when subtracting a constant? To answer this question, let’s look at an example.

Consider the function [latex]f(x)=|x+3|[/latex] and evaluate this function at [latex]x-3.[/latex] Since [latex]f(x-3)=|x|[/latex] and [latex]x-3<x[/latex], the graph of [latex]f(x)=|x+3|[/latex] is the graph of [latex]y=|x|[/latex] shifted left 3 units. Similarly, the graph of [latex]f(x)=|x-3|[/latex] is the graph of [latex]y=|x|[/latex] shifted right 3 units (Figure 16).

Vertical Scaling (Stretched/Compressed)

A vertical scaling of a graph occurs if we multiply all outputs [latex]y[/latex] of a function by the same positive constant. For [latex]c>0[/latex], the graph of the function [latex]cf(x)[/latex] is the graph of [latex]f(x)[/latex] scaled vertically by a factor of [latex]c[/latex]. If [latex]c>1[/latex], the values of the outputs for the function [latex]cf(x)[/latex] are larger than the values of the outputs for the function [latex]f(x)[/latex]; therefore, the graph has been stretched vertically. If [latex]0<c<1[/latex], then the outputs of the function [latex]cf(x)[/latex] are smaller, so the graph has been compressed. For example, the graph of the function [latex]f(x)=3x^2[/latex] is the graph of [latex]y=x^2[/latex] stretched vertically by a factor of 3, whereas the graph of [latex]f(x)=\frac{x^2}{3}[/latex] is the graph of [latex]y=x^2[/latex] compressed vertically by a factor of 3 (Figure 17).

Horizontal Scaling (Stretched/Compressed)

The horizontal scaling of a function occurs if we multiply the inputs [latex]x[/latex] by the same positive constant. For [latex]c>0[/latex], the graph of the function [latex]f(cx)[/latex] is the graph of [latex]f(x)[/latex] scaled horizontally by a factor of [latex]c[/latex]. If [latex]c>1[/latex], the graph of [latex]f(cx)[/latex] is the graph of [latex]f(x)[/latex] compressed horizontally. If [latex]0<c<1[/latex], the graph of [latex]f(cx)[/latex] is the graph of [latex]f(x)[/latex] stretched horizontally. For example, consider the function [latex]f(x)=\sqrt{2x}[/latex] and evaluate [latex]f[/latex] at [latex]\dfrac{x}{2}.[/latex] Since [latex]f(\frac{x}{2})=\sqrt{x}[/latex], the graph of [latex]f(x)=\sqrt{2x}[/latex] is the graph of [latex]y=\sqrt{x}[/latex] compressed horizontally. The graph of [latex]y=\sqrt{\frac{x}{2}}[/latex] is a horizontal stretch of the graph of [latex]y=\sqrt{x}[/latex] (Figure 18).

Reflection

We have explored what happens to the graph of a function [latex]f[/latex] when we multiply [latex]f[/latex] by a constant [latex]c>0[/latex] to get a new function [latex]cf(x)[/latex]. We have also discussed what happens to the graph of a function [latex]f[/latex] when we multiply the independent variable [latex]x[/latex] by [latex]c>0[/latex] to get a new function [latex]f(cx)[/latex]. However, we have not addressed what happens to the graph of the function if the constant [latex]c[/latex] is negative. If we have a constant [latex]c<0[/latex], we can write [latex]c[/latex] as a positive number multiplied by [latex]-1[/latex]; but, what kind of transformation do we get when we multiply the function or its argument by [latex]-1[/latex]? When we multiply all the outputs by [latex]-1[/latex], we get a reflection about the [latex]x[/latex]-axis. When we multiply all inputs by [latex]-1[/latex], we get a reflection about the [latex]y[/latex]-axis. For example, the graph of [latex]f(x)=−(x^3+1)[/latex] is the graph of [latex]y=(x^3+1)[/latex] reflected about the [latex]x[/latex]-axis. The graph of [latex]f(x)=(−x)^3+1[/latex] is the graph of [latex]y=x^3+1[/latex] reflected about the [latex]y[/latex]-axis (Figure 19).

Multiple Transformations

If the graph of a function consists of more than one transformation of another graph, it is important to transform the graph in the correct order. Given a function [latex]f(x)[/latex], the graph of the related function [latex]y=cf(a(x+b))+d[/latex] can be obtained from the graph of [latex]y=f(x)[/latex] by performing the transformations in the following order.

1. Horizontal shift of the graph of [latex]y=f(x)[/latex]. If [latex]b>0[/latex], shift left. If [latex]b<0[/latex], shift right.
2. Horizontal scaling of the graph of [latex]y=f(x+b)[/latex] by a factor of [latex]|a|[/latex]. If [latex]a<0[/latex], reflect the graph about the [latex]y[/latex]-axis.
3. Vertical scaling of the graph of [latex]y=f(a(x+b))[/latex] by a factor of [latex]|c|[/latex]. If [latex]c<0[/latex], reflect the graph about the [latex]x[/latex]-axis.
4. Vertical shift of the graph of [latex]y=cf(a(x+b))[/latex]. If [latex]d>0[/latex], shift up. If [latex]d<0[/latex], shift down.

We can summarize the different transformations and their related effects on the graph of a function in the following table.

Watch the following video to see the worked solution to Example: Transforming a Function